Enzyme Kinetics

Enzyme:

* biological catalyst which speeds up rate of chemical reactions
* specific for particular substrates
* can regulate reactions by responding to signals from surroundings
* usually made of protein, but some RNA molecules also can act as catalysts

Kinetics

* study of reaction rates

rate of any chemical reaction limited by energy needed to reach transition state

equilibrium ratio of A/B dependent on relative free energy of A and B (not changed by enzyme)

Enzyme speeds up rate of reaction:

* stabilizes transition state - lower energy required to achieve it
* also:
* entropy reduction - reduces disorder by orienting substrates for reaction
* covalent catalysis - may form covalent bond to substrate during reaction
--> new pathway of reaction with lower activation energy (also important for reaction coupling)
* may bind metal of other cofactor that does work of reaction (Mg++)

Enzyme binds reactants (substrates - S) at active site
* active site typically a small portion of the volume of the enzyme,
* rest of enzyme necessary for structural stability, proper folding, also may have regulatory domains

Substrate bound through series of weak interactions:
Hbonds, VDW, hydrophobics
--> specificity (substrate missing any key feature is not bound efficiently)

How does S fit E? Lock-and-key? Actually more of an induced fit, enzyme changes conformation upon binding substrate

An example of an enzyme:
Chymotrypsin-
* a serine protease (i. e. has serine at active site)
* digests proteins at C-term side of Y, W or F residues
* important active site residues: Asp(D)102, His(H)57 and Ser(S)195.
* Uses stabilization of transition state, entropy reduction, and covalent catalysis to enhance reaction rate

rate of uncatalyzed reaction: 10^-10/sec
rate of uncatalyzed reaction: 10²/sec
rate enhancement of: 10¹²/sec!

Serine Protease Mechanism

Enzyme binds substrate, orienting it for subsequent reaction. (Reduction of entropy)

His57 accepts proton from ser195, allowing O of ser to attack C=O carbon of substrate. Protonated his stabilized by negative charge of neighboring asp102

Tetrahedral transition state intermediate formed, stabilized by H-bonds from backbone NH protons of residues 193 and 195. Enzyme is covalently linked to substrate. Peptide N of substrate accepts proton from his57.

Electrons move from C-N bond to +-charged N, breaking C-N bond of substrate. C=O bond reformed.

Amino fragment of substrate released, carboxyl fragment remains covalently bound to ser195 as acyl-enzyme intermediate.

His57 accepts proton from solvent H2O, remaining OH attacks C=O carbon of acyl-enzyme intermediate.

Tetrahedral transition state formed again, stabilized by H-bonds (not shown this time). Subsequently, acyl bond breaks, proton transferrred from his57 to ser195, C=O bond reformed on product.

Carboxyl proton lost.

C-terminal product released.

Measuring rate of enzyme catalysis - enzyme kinetics

(early in reaction [P]<<[S])

S=substrate, P=product, E= free enzyme, ES=enzyme-substrate complex E_T=total enzyme

[ES] ~ constant : steady state

Varying the substrate concentration changes the rate of reaction:

The rate of product formation, V:
V = k₃ [ES]
and the maximum reaction rate, V_max , occurs when all enzyme is bound to substrate:
V_max = k₃ [E_T]

Under steady-state conditions:

rate of ES formation =rate of ES breakdown

[E][S]k₁ = [ES]k₂+[ES]k₃
or:
[E][S]k₁ = [ES] (k₂+ k₃)

solving for [ES]:

since k₁,k₂, and k₃ should be constant for a particular E and S, we'll define a constant K_M , based upon them:

substituting into the previous equation:

we don't really know [E] or [ES], but:
[E]=[E_T]-[ES]
substituting for [E]:

factoring:
[E_T] [S]-[ES][S] = [ES]K_M

rearranging:
[E_T] [S] = [ES][S] + K_M [ES]

factoring:
[E_T] [S] = [ES] ([S] + K_M )

dividing both sides by ([S] + K_M ):

now substitute this into the equation for reaction velocity:
V = k₃ [ES]:

when the enzyme is fully saturated, velocity reaches V_max :
V_max = k₃ [E_T]
substituting again:

This is the Michaelis-Menten equation.

what happens when [S] = K_M ?

or:
V = V_max /2
Reaction velocity is one-half of V_max when [S] = K_M .

But it is hard to determine V_max and K_M from this type of graph. How do you actually determine V_max and K_M ?
You can use a computer graphing program to fit the data to a curve.
Or you can rearrange the Michaelis-Menten equation to make it easier to plot and determine V_max and K_M.

Start with the Michaelis-Menten equation:

take the inverse of both sides:

factor:

which can be rewritten as:

compare to the plot of a linear equation:

So plotting 1/S for x and 1/V for y yields a straight line with 1/ V_max for the y-intercept (b). The slope (m) = K_M / V_max and the x-intercept= -1/ K_M .
This is the Lineweaver-Burk plot:

Importance of K_M and k_cat:

In vivo often [S] ~ K_M:
if [S] = 1000 K_M, rate still only 2X higher than when [S] = K_M (waste of S; reaction rate is insensitive to [S])
if [S] = K_M /1000, most of E unoccupied (waste of E)
Knowing K_M useful for setting up reactions in lab:
eg. picking a DNA polymerase for 32P labeling (expensive, dangerous in large quantities)
E. coli polI K_M for dNTPs ~1uM
AMV reverse transcriptase K_M ~1mM

So polI can more efficiently incorporate small [ ] of dNTP

polI formerly used for DNA sequencing, but K_M varies for each dNTP and ddNTP, so optimizing reaction was quite tedious. T7, Taq polymerase: similar K_M for all dNTP and ddNTPs easier to optimize

k₃ (k_cat for multistep reaction)

V_max = k₃ [E_T]
k₃= reactions catalyzed/mol enzyme-sec
each reaction requires 1/ k₃ sec to occur

carbonic anhydrase: k₃ =6X10⁵/s time per reaction =1.7 us
chymotrypsin: k_cat =1X10²/s time per reaction =0.01 s
ribonuclease P =0.2/s time per reaction =5 s
k_cat/ K_M

when [S] << K_M

so V is dependent upon k_cat/ K_M at a given [E_T] [S].

Can compare affinities for different substrates using low [S]:

Chymotrypsin:

showing preference for bulky hydrophobics at active site.

if k₃>> k₂

k₁ limited by diffusion of S to E: 10⁸ -10⁹ /M-sec

"catalytic perfection" production of product limited by rate of encountering substrate in solution.

carbonic anhydrase, acetyl cholinesterase - ~ catalytically perfect, k_cat/ K_M around 10⁸ /M-sec

Even this limit can be surpassed by enzymatic assemblies - related enzymes co-localized, product of one E released to next to use as S. Faster than diffusion.

Multicomponent reactions

Most reactions more complex than

such as:

Kinetics also become much more complicated

Do reactants bind to enzyme at same time? (ternary complex formation)

Do reactants bind in particular order?

Random binding order:

ordered binding

Ping-pong Bi-Bi:

Equations can get pretty hairy, usually keep one [S] constant and high, vary the other to determine kinetic parameters.

Inhibitors

Specific compounds that slow enzymatic reactions

Very important tool for studying reaction components

Major types of competitors:

Competitive inhibitors:

Usually resemble substrate (including product)

Inhibitor (I) binds to active site instead of substrate

V_max not changed, just requires more S to reach it. Apparent K_M , K_MAPP goes up.

Noncompetitive inhibition

I and S bind to different sites, but only ES --> P

K_M not affected, but V_max is decreased.